Photon energy redshift from a tensor metric perturbation ($d\log ϵ/d\eta =-{\dot{h}}_{ij}{e}^i{e}^j/2$)

For a tensor cosmological perturbation $h_{ij}$, the time geodesic equation in a local orthonormal tetrad gives $d\log ϵ/d\eta =-{\dot{h}}_{ij}{e}^i{e}^j/2$. The photon comoving energy removes the background redshift; the dot is a partial time derivative, whereas the energy derivative follows the ray.